Question

How many ways can three integers be selected from \(3 n\) consecutive integers so that the sum is a multiple of \(3 ?\) Here, \(n\) is a positive integer. What if the three chosen integers must be distinct?

Short answer

Without distinct condition: \(3\binom{n}{3} + n^3\) ways; with distinct: \(n^3\) ways.

Step-by-step solution

Understanding the Problem

We need to determine how many ways three integers can be chosen from a sequence of \(3n\) consecutive integers such that their sum is a multiple of 3. In addition, we want to find out how many such sets can be formed if the integers must be distinct.

Sequence Analysis

Consider the sequence of \(3n\) consecutive integers: \(a, a+1, a+2, \ldots, a+3n-1\). Each integer \(x\) from this sequence has a remainder of 0, 1, or 2 when divided by 3, often referred to as the residue of \(x\).

Residue Analysis

In any group of 3 consecutive integers, there is exactly one integer with each possible residue (0, 1, or 2) modulo 3. Therefore, in a sequence of \(3n\) integers, there are \(n\) integers with residue 0, \(n\) integers with residue 1, and \(n\) integers with residue 2.

Selecting Residues

To have the sum of three integers add up to a multiple of 3, their residues should add to a multiple of 3. Possible combinations are: (0,0,0), (1,1,1), (2,2,2), and one integer each from (0,1,2).

Calculating Combinations

- For (0,0,0), we have \(\binom{n}{3}\) ways to choose 3 integers all with residue 0.- Similarly, \(\binom{n}{3}\) ways for (1,1,1) and for (2,2,2).- For (0,1,2), we have \(n \times n \times n = n^3\) ways, as we select one residue from each type.

Total Combinations Without Distinct Constraint

Adding these up gives us the total number of ways to pick the integers: \( \binom{n}{3} + \binom{n}{3} + \binom{n}{3} + n^3 = 3\binom{n}{3} + n^3 \).

Distinctness Condition

When choosing distinct integers, (0,0,0), (1,1,1), and (2,2,2) are not possible because they would mean selecting the same number. Thus, only (0,1,2) remains viable, which remains unchanged as each is distinct already.

Final Calculation With Distinctness

With the distinct condition, only the 0,1,2 combination contributes, giving us \(n^3\) ways to choose the integers.

Key concepts

Consecutive Integers

Consecutive integers are numbers that follow each other without any gaps. For example, the numbers 1, 2, and 3 are consecutive integers. They come directly one after the other on the number line.
In mathematical problems, we often deal with consecutive integers because they form a simple, predictable pattern. This pattern can be quite useful when determining properties that help with calculations, such as sums or differences.
When dealing with a sequence of consecutive integers, like in our problem with \(3n\) integers, each number can easily be related to the others by adding or subtracting a fixed number.

Multiple of Three

A number is a multiple of three if it can be divided by 3 without any remainder. For example, the numbers 3, 6, and 9 are all multiples of three because 3 divides each of them exactly.
Finding multiples of three among selected integers involves understanding their sum. If the sum of several integers is also a multiple of three, each component of the sum can sometimes offer clues. Like solving a puzzle, this concept uses basic division rules to determine relationships between numbers.
In our exercise, figuring out ways to sum integers from a sequence to be a multiple of three can lead to valuable conclusions about the possible selections.

Residues Modulo

Residues modulo, often called just 'mod', refer to the remainder when one number is divided by another. When we say 'residues modulo 3', we're interested in the remainders of numbers after dividing them by 3. This helps categorize numbers into three groups: those with residue 0, 1, or 2.
Understanding residues as a concept is central to solving our problem because it organizes the integers into meaningful patterns. For instance, within any consecutive set of 3 numbers such as (1, 2, 3), one will have a residue of 0, another 1, and another 2.
This predictable organization helps simplify the search for sets of integers that collectively meet the condition of their sum being a multiple of 3.

Distinct Integers

Distinct integers are those that are different from one another. In contexts like our exercise, it means no integers in the selection pool can be the same.
This distinction impacts how we calculate possible selections. In terms of residues, choosing three integers with the same residue, such as three 0s (0,0,0), would inherently be non-distinct if they are from a group of consecutive integers, since they would all represent the same number when actually picked.
Instead, we focus on combinations like (0,1,2) where each number is different, fulfilling both the distinct requirement and the condition for the sum to be a multiple of 3. Here, the distinctness criteria narrow down the choices significantly, highlighting the importance of evaluating each potential combination carefully.